QSPR Analysis of certain Distance Based Topological Indices

Shailaja Shirakol, Manjula Kalyanshetti and Sunilkumar M. Hosamani

Abstract

In QSAR/QSPR study, topological indices are utilized to guess the bioactivity of chemical compounds. In this paper, we study the QSPR analysis of selected distance and degree-distance based topological indices. Our study reveals some important results which help us to characterize the useful topological indices based on their predicting power.

1 Introduction

The molecular descriptor is the final result of logic and mathematical procedure which transform chemical information encoded within a symbolic representation of a molecule into a useful member or the result of some standardized experiments. Attention is paid to the term "useful" with its double meanings. It means that the number can give more insights into the interpretation of the molecular properties and / or is able to take part in a model for the prediction of some interesting property of the molecules.

A fundamental concept of chemistry is that the structural characteristics of a molecule are responsible for its properties. Topological indices are a convenient means of translating chemical constitution into numerical values which can be used for correlation with physical properties in quantitative structure-property/activity relationship (QSPR/QSAR) studies. The use of graph invariant (topological indices) in QSPR and QSAR studies has become of major interest in recent years. Topological indices have found application in various areas of chemistry, physics, mathematics, informatics, biology, etc [1, 7, 26], but their most important use to date is in the non-empirical Quantitative Structure- Property Relationships (QSPR) and Quantitative Structure -Activity Relationships (QSAR) [5, 14, 19, 21, 23, 24, 27].

2 Survey of Selected Distance and Degree-Distance Based Topological Indices

  1. Wiener Index: The Wiener index is named after Harry Wiener, who introduced it in 1947; at the time, Wiener called it the "path number"[24]. It is the oldest topological index related to molecular branching. Based on its success, many other topological indices of chemical graphs [2,3,13], based on information in the distance matrix of the graph, have been developed subsequently to Wiener’s work. Which is defined as:let G be any connected graph of order n and size m. Then Wiener index of G is denoted by W(G) and is defined as follows.
    W(G)=12uV(G)dG(u,v)
  2. Terminal Wiener Index: The concept of terminal Wiener index was put forward by Petrović et el. [9] some-what later but independently, Szekely et al. [25] arrived at the same idea. If G has k-pendent vertices labeled by v1;v2. . .vk , then its terminal distance matrix is the square matrix of order k whose (i, j)-th entry is d(vi,vj \G). Terminal distance matrices were used for modeling amino acid sequences of proteins and of the genetic code [12, 17, 18].The terminal Wiener index TW(G) of a connected graph G is defined as the sum of the distances between all pairs of its pendent vertices.Thus if VT = {v1;v2; . . . ,vk} is the set of all pendent vertices of G, then
    TW(G)={u,vVT(G)}d(u,v\G)=1i<jkd(u,v\G)
  3. Degree Distance Index: The degree distance was introduced by Dobrynin and Kochetova [5] as a weighted version of the Wiener index. The degree distance of G, denoted by DD(G), is defined as follows
    DD(G)={u,v}V(G)dG(u,v)[degG(u)+degG(v)].
  4. Gutman Index: The Gutman index was put forward in [10] as a multiplicative version of degree-distance index which is defined as follows.
    GI(G)={u,v}V(G)dG(u,v)[degG(u)degG(v)].
  5. Ashwini Index: Motivated by the terminal Wiener index Hosamani [15] has introduced a novel topological index viz, Ashwini index of a molecular graph G. Which is based on the terminal distance between any pair of pendant vertices together with their neighborhood degrees.
    AT=1i<jndTvi,vjdegTNui+degTNvj.
    Where N(v) = {u ∈ V(G) : uv ∈ E(G)}.
  6. SM- Index: Motivated by the Gutman index and Ashwini index of a molecular graph G, we define here a new topological invariant namely SM-index of a molecular graph G. Which is defined as follows:
    SM(T)=1i<jndT(vi,vj)[degT(N(ui))degTN(vj)].
    Where N(v) = {uV(G) : uvE(G)}.
  7. Hyper Wiener Index: In 1993, Milan Randić [20] introduced a distance based quantity, he named it as hyper Wiener index and denoted by WW. His definition could be applied only to trees, and was in possible to use for cycle-containing graphs.
    WW(G)=12{u,v}V(G)[dG(u,v)+dG2(u,v)]
    which could be applied to all connected graphs, since then the above formula is used the definition of the hyper Wiener index.

3 The Use of Selected Distance and Degree-Distance Based Topological Indices in QSPR Studies

We have used three distance based topological indices and four degree-distance based topological indices viz,Wiener Index (W(G)),Terminal Wiener Index (TW(G)),Hyper Wiener Index (WW(G)) [distance baesd TI’s] and Degree-distance Index, Gutman Index (GI(G)),Ashwini Index AG,SNM-Index [degree-distance based TI’s] respectively for modeling eight representative physical properties [boiling points(BP), molar volumes (mv) at 20°C, molar refractions (mr) at 20°C, heats of vaporization (hv) at 25°C, surface tensions (st) 20°C and melting points (mp)] of the 70 alkanes from n-butanes to nonanes. Values for these property were taken from Dejan Plavsić et. al [16]. The above said Distance and Degree-Distance topological indices and the experimental values for the physical properties of 70 alkanes are listed in Table 1 and 2 respectively.

Table 1
S.No.Alkanebp(°C)mv(cm3)mr (cm3)hv(kJ)ct(°C)cp(atm)st(dyne/cm)mp(°C)
1Butane-0.500152.0137.47-138.35
22-methyl propane-11.730134.9836-159.60
3Pentane36.074115.20525.265626.42196.6233.3116.00-129.72
42-methyl butane27.852116.42625.292324.59187.7032.915.00-159.90
52,2 dimethylpropane9.503112.07425.724321.78160.6031.57-16.55
6Hexane68.740130.68829.906631.55234.7029.9218.42-95.35
72-methylpentane60.271131.93329.945929.86224.9029.9517.38-153.67
83-methyalpentane63.282129.71729.801630.27231.2030.8318.12-118.00
92,2-methylbutane49.741132.74429.934727.69216.2030.6716.30-99.87
102,3-dimethylbutane57.988130.24029.810429.12227.1030.9917.37-128.54
11Heptanes98.427146.54034.550436.55267.5527.0120.26-90.61
122-methylhexane90.052147.65634.590834.80257.9027.219.29-118.28
133-methylhexane91.850145.82134.459735.08262.4028.119.79-119.40
143-ethylpentane93.475143.51734.282735.22267.6028.620.44-118.60
152,2-dimethylpentane79.197148.69534.616632.43247.7028.418.02-123.81
162,3 -dimethylpentane89.784144.15334.323734.24264.6029.219.96-119.10
172,4-dimethylpentane80.500148.94934.619232.88247.1027.418.15-119.24
183,3-dimethylpentane86.064144.53034.332333.02263.003019.59-134.46
19Octane125.665162.59239.192241.48296.2024.6421.76-56.79
202-methylheptane117.647163.66339.231639.68288.0024.820.60-109.04
213 -methylheptane118.925161.83239.100139.83292.0025.621.17-120.50
224-methylheptane117.709162.10539.117439.67290.0025.621.00-120.95
233-ethylhexane118.53160.0738.9439.40292.0025.7421.51
242,2-dimethylhexane10.84164.2839.2537.29279.0025.619.60-121.18
252,3-dimethylhexane115.607160.3938.9838.79293.0026.620.99
262,4-dimethylhexane109.42163.0939.1337.76282.0025.820.05-137.50
272,5-dimethylhexane109.10164.6939.2537.86279.002519.73-91.20
283,3 -dimethy lhexane111.96160.8739.0037.93290.8427.220.63-126.10
293,4-dimethy lhexane117.72158.8138.8439.02298.0027.421.64
303 -ethyl-2-methylpentane115.65158.7938.8338.52295.0027.421.52-114.96
313-ethyl-3-methylpentane118.25157.0238.7137.99305.0028.921.99-90.87
322,2,3-trimethylpentane109.84159.5238.9236.91294.0028.220.67-112.27
332,2,4-trimethylpentane99.23165.0839.2635.13271.1525.518.77-107.38
342,3,3-trimethylpentane114.76157.2938.7637.22303.002921.56-100.70
352,3,4-trimethylpentane113.46158.8538.8637.61295.0027.621.14-109.21
36Nonane150.79178.7143.8446.44322.0022.7422.92-53.52
372-methyloctane143.26179.7743.8744.65315.0023.621.88-80.40
383-methyloctane144.18177.9543.7244.75318.0023.722.34-107.64
394-methyloctane142.48178.1543.7644.75318.3023.0622.34-113.20
403-ethylheptane143.00176.4143.6444.81318.0023.9822.81-114.90
414-ethylheptane1141.20175.6843.4944.81318.3023.9822.81
422,2-dimethylheptane132.69180.5043.9142.28302.0022.820.80-113.00
432,3-dimethylheptane140.50176.6543.6343.79315.0023.7922.34-116.00
442,4-dimethylheptane133.50179.1243.7342.87306.0022.723.30
452,5-dimethylheptane136.00179.3743.8443.87307.8022.721.30
462,6- dimethylheptane135.21180.9143.9242.82306.0023.720.83-102.90
473,3- dimethylheptane137.300176.89743.687042.66314.0024.1922.01
483,4- dimethylheptane140.600175.34943.547343.84322.7024.7722.80
493,5- dimethylheptane136.000177.38643.637942.98312.3023.5921.77
504,4- dimethylheptane135.200176.89743.602242.66317.8024.1822.01
513-ethyl-2-methylhexane138.000175.44543.655043.84322.7024.7722.80
524-ethyl-2-methylhexane133.800177.38643.647242.98330.3025.5621.77
533 -ethyl- 3-methylhexane140.600173.07743.268044.04327.2025.6623.22
542,2,4- trimethylhexane126.540179.22043.763840.57301.0023.3920.51-120.00
552,2,5- trimethylhexane124.084181.34643.935640.17296.6022.4120.04-105.78
562,3,3- trimethylhexane137.680173.78043.434742.23326.1025.5622.41-116.80
572,3,4- trimethylhexane139.000173.49843.491742.93324.2025.4622.80
582,3,5- trimethylhexane131.340177.65643.647441.42309.4023.4921.27-127.80
593,3,4- trimethylhexane140.460172.05543.340742.28330.6026.4523.27-101.20
603,3-diethylpentane146.168170.18543.113443.36342.8026.9423.75-33.11
612,2-dimethyl-3-ethylpentane133.830174.53743.457142.02322.6025.9622.38-99.20
622,3-dimethyl-3-ethylpentane142.000170.09342.954242.55338.6026.9423.87
632,4-dimethyl-3-ethylpentane136.730173.80443.403742.93324.2025.4622.80-122.20
642,2,3,3-tetramethylpentane140.274 169.49543.214741.00334.5027.0423.38-99.0
652,2,3,4- tetramethylpentane133.016173.55743.435941.00319.6025.6621.98-121.09
662,2,4,4- tetramethylpentane122.284178.25643.874738.10301.6024.5820.37-66.54
672,3,3,4- tetramethylpentane141.551169.92843.201641.75334.5026.8523.31-102.12
Table 2
S.No.AlkaneW(G)TW{G)DD{G)G1{G)A(G)SNM{G)HW{G)
1Butane1032819121246
22-methyl propane962013365427
3Pentane20860441616146
42-methyl butane1885236425490
52,2 dimethylpropane16844289619252
6Hexane355110822020371
72-methylpentane321096735266254
83-methyalpentane311094694652217
92,2-methylbutane26158257102168142
102,3-dimethylbutane2916846196144161
11Heptanes5661821462424812
122-methylhexane52121661306278604
133-methylhexane50121611225562506
153-ethylpentane48121501444848408
162,2-dimethylpentane4916142106120192370
172,3 -dimethylpentane4615154109108144352
182,4-dimethylpentane4816150114120180426
193,3-dimethylpentane441413698104144296
20Octane84728023128281596
212-methylheptane791426321172901261
223 -methylheptane761424821264721072
234-methylheptane751422419364721011
243-ethylhexane72142321905656822
252,2-dimethylhexane7121228179138216845
262,3 -dimethylhexane7122291275118156766
272,4-dimethylhexane7123231179125168803
282,5 -dimethylhexane7424240191144216962
293,3-dimethylhexane6721212163113150649
303,4-dimethylhexane6822216168108131668
313-ethyl-2-methylpentane6722215163108130607
323-ethyl-3-methylpentane6421239129102120514
332,2,3 -trimethylpentane6327196147198318495
342,2,4-trimethylpentane6632194152228330606
352,3,3-trimethylpentane6027242200194314458
362,3,4-trimethylpentane6532198155192288551
37Nonane120842236532322892
382-methyloctane11416391327821022388
393-methyloctane1101631026873822076
404-methyloctane1081635329573821920
413-ethylheptane1051633127172721604
424-ethylheptane1021623919068681452
432,2-dimethylheptane104243433781562401718
442,3-dimethylheptane102253362641331741548
452,4-dimethylheptane102263422681481901524
462,5-dimethylheptane110273442691471981646
472,6- dimethylheptane108283442801682521926
483,3- dimethylheptane98243202771381841340
493,4- dimethylheptane98253203921221471298
503,5- dimethylheptane100263202641281561396
514,4- dimethylheptane96243142481361841218
523-ethyl-2-methylhexane96253152481221461146
534-ethyl-2-methylhexane98263032491261481244
543 -ethyl- 3-methylhexane9224295228114130992
552,2,4- trimethylhexane94363012382373781108
562,2,5- trimethylhexane98383222702704741328
572,3,3- trimethylhexane9032289256210318936
582,3,4- trimethylhexane9236295238199273992
592,3,5- trimethylhexane96383172512283421188
603,3,4- trimethylhexane8734278214201278838
613,3-diethylpentane8824134969696796
622,2-dimethyl-3-ethylpentane8832279213208304814
632,3-dimethyl-3-ethylpentane8634266197186250740
642,4-dimethyl-3-ethylpentane9036291224200276870
652,2,3,3-tetramethylpentane8244223170316560628
662,2,3,4- tetramethylpentane8647273209327564758
672,2,4,4- tetramethylpentane8640280216316530850
682,3,3,4- tetramethylpentane844726420028464729

4 Regression Models

We have tested the following linear regression model

P=A+B(TI)

where P = physical property, TI = topological index.

Using (3.1), we have obtained the following different linear models for each degree based topological index, which are listed below.

  1. Wiener indexW(G):
    bp=20.8432+[W(G)]1.2203
    mv=113.868+[W(G)]0.6412
    mr=24.7432+[W(G)]0.1944
    hv=23.710+[W(G)]0.1998
    ct=179.262+[W(G)]1.4606
    cp=34.1143[W(G)]0.1025
    st=16.126748+[W(G)]0.0634
    mp=129.02+[W(G)]0.2909
  2. Terminal wiener indexTW(G):
    bp=68.2472+[TW(G)]1.9072
    mv=139.62+[TW(G)]1.002
    mr=32.25+[TW(G)]0.3178
    hv=37.616+[TW(G)]0.055
    ct=230.099+[TW(G)]2.5196
    cp=29.5512[TW(G)]0.1327
    st=19.1536+[TW(G)]0.088
    mp=579.080[TW(G)]32.24
  3. Hyper wiener indexHW(G):
    bp=69.8172+[HW(G)]0.04586
    mv=140.3322+[HW(G)]0.02365
    mr=33.2598+[HW(G)]0.00665
    hv=31.82212+[HW(G)]0.007513
    ct=245.468+[HW(G)]0.04589
    cp=30.4719[HW(G)]0.00433
    st=19.1519+[HW(G)]0.001937
    mp=120.603+[HW(G)]0.013596
  4. Degree distance indexDD(G):
    bp=32.2663+[DD(G)]0.3432
    mv=119.529+[DD(G)]0.182
    mr=37.18+[DD(G)]0.00714
    hv=37.144+[DD(G)]0.00732
    ct=196.428+[DD(G)]0.3932
    cp=33.418[DD(G)]0.0298
    st=17.121+[DD(G)]0.0162
    mp=122.787+[DD(G)]0.0641
  5. Gutman indexGI(G):
    bp=38.62+[GI(G)]0.3983
    mv=201.965[GI(G)]0.20958
    mr=28.02+[GI(G)]0.061
    hv=26.978+[GI(G)]0.0632
    ct=203.4429+[GI(G)].4577
    cp=20.218[GI(G)]0.034733
    st=17.37+[GI(G)]0.0190
    mp=119.40+[GI(G)]0.0616
  6. Ashwini indexA(G):
    bp=135.6962+AG0.1870
    mv=148.45+AG0.1095
    mr=35.573+AG0.034
    hv=39.95+AG0.00828
    ct=237.8298+AG0.3766
    cp=28.4180AG0.01427
    st=19.970+AG0.0078
    mp=113.0059+AG0.03
  7. SM indexSM(G):
    bp=97.968+[SNM(G)]0.069
    mv=74.45+[SNM(G)]0.46
    mr=10.394+[SNM(G)]0.152
    hv=37.749+[SNM(G)]0.00587
    ct=268.506+[SNM(G)]0.0956
    cp=27.68[SNM(G)]0.00582
    st=20.486+[SNM(G)]0.00266
    mp=109.163[SNM(G)]0.00016

5 Discussion and Concluding Remarks

By inspection of the data given in tables 3 to 9, It is possible to draw numbers of conclusion for the given distance and degree-distance based TIs.

Table 3

Statical parameters for the linear QSPR model for Wiener index.

Physical PropertiesNabrsF
Boiling point7020.84321.22030.92114.3278388.436
Molar volume67113.8680.64120.9704.320841037.804
Molar refraction6724.74320.19440.9621.45880795.781
Heats of vaporization6723.7100.19980.9641.45328846.841
Critical temperature70179.2621.46060.89919.8772285.433
Critical Pressure7034.1143-0.10250.9211.2167380.698
Surface tension6616.1267480.06340.8151.14383126.812
Melting point52-129.020.29090.31725.875375.585
Table 4

Statical parameters for the linear QSPR model for terminal Wiener index.

Physical PropertiesNabrsF
Boiling point7068.24721.90720.57430.381533.333
Molar volume67139.621.0020.60614.1719437.698
Molar refraction6732.250.31780.6444.0610146.075
Heats of vaporization6737.6160.0550.4434.8787615.910
Critical temperature70230.0992.51960.62035.566542.39
Critical Pressure7029.5512-0.13270.4752.750319.816
Surface tension6619.15360.0880.4811.7310319.315
Melting point52579.080-32.2140.09627.15320.461
Table 5

Statical parameters for the linear QSPR model for hyper Wiener index.

Physical PropertiesNabrsF
Boiling point7069.81720.045860.72225.675873.879
Molar volume67140.33220.023650.75911.5915488.511
Molar refraction6733.25980.0019370.5551.6434628.430
Heats of vaporization6731.822120.0075130.7893.34131107.499
Critical temperature70245.4680.045890.8091.1358129.100
Critical Pressure7030.4719-0.004330.63634.98726.078
Surface tension6619.15190.00190.5551.6434628.430
Melting point52-120.6030.0135960.31825.865145.629
Table 6

Statical parameters for the linear QSPR model for Degree distance index.

Physical PropertiesNabrsF
Boiling point7032.26630.34320.86318.7367198.428
Molar volume67119.5290.1820.9197.03409315.874
Molar refraction6737.810.007140.8972.34677267.614
Heats of vaporization6737.1440.007320.8972.40325268.443
Critical temperature70196.4280.39320.80926.6334128.863
Critical Pressure7033.418-0.02980.8901.4243259.425
Surface tension6617.1210.01620.7161.3795667.174
Melting point52-122.7870.06410.23226.539272.839
Table 7

Statical parameters for the linear QSPR model for Gutman index.

Physical PropertiesNabrsF
Boiling point7038.620.39830.85819.0533189.648
Molar volume67201.965-0.209580.8928.05741252.709
Molar refraction6728.020.0610.8742.58307209.543
Heats of vaporization6726.9780.06320.8822.56979226.624
Critical temperature70203.44290.45770.80726.7911126.553
Critical Pressure7020.218-0.0347330.8891.4332255.359
Surface tension6617.370.01900.7111.3885365.486
Melting point52-119.400.06160.18726.800081.815
Table 8

Statical parameters for the linear QSPR model for Ashwini index.

Physical PropertiesNabrsF
Boiling point70135.6962-0.18700.42033.64414.578
Molar volume67148.450.10950.49215.5133420.706
Molar refraction6735.5730.0340.5194.5380823.949
Heats of vaporization6739.95-0.008230.2885.212355.885
Critical temperature70237.82980.37660.45940.259618.155
Critical Pressure7023.4180-0.014270.3812.890111.523
Surface tension6619.9700.00780.3201.871217.299
Melting point52-113.00590.030.5614.9073122.913
Table 9

Statical parameters for the linear QSPR model for SM index.

Physical PropertiesNabrsF
Boiling point7097.9680.0690.29235.47376.3828
Molar volume6774.470.460.37816.494910.809
Molar refraction6710.3940.1520.4004.886612.352
Heats of vaporization6737.7490.005870.1585.375071.658
Critical temperature70268.5060.09560.32642.83878.093
Critical Pressure7027.68-0.005820.2882.99316.145
Surface tension6620.4860.002660.1981.935902.6614
Melting point52-109.163-0.000160.3395.12088.7442

First, the famous and much studied distance based Topological index viz, Wiener index found to be more suitable tool to predict the physical properties of alkanes.The Wiener Index shows good correlation with almost all physical properties of alkanes which are listed in table 3 except molar volume and surface tension of alkanes.The correlation coefficient value lies between 0.815 to 0.970. The QSPR study reveals that Wiener Index is more suitable to predict heats of vapourization and molar volumes of alkanes with correlation coefficient value r=0.964, and r = 0.0.970 respectively.

In addition the result for Terminal Wiener index revealed that the recent advocated idea of using Terminal Wiener index did not pass the test. This important details seems to have ignored in recent paper [9], on Terminal Wiener index.

Recently introduced distance based topological invariant viz, Hyper Wiener index found to be adequate for any structure-property correlation, except for critical temperatures of alkanes with correlation coefficient value r=0.809.

The QSPR study of degree-distance index in tables 6 reveals that the degree-distance index is an useful topological invariant. It shows good correlation with almost all physical properties which are listed in Table 6, except surface tension and melting points of alkanes. The correlation coefficient values lies between 0. 809 to 0.919. The degree-distance index is more suitable to predict the molar volume and heats of vaporization with r = 0 :919 and r = 0 :897 respectively.

The multiplicative version of degree-distance index is known as Gutman Index. By observing the results in table 7,One can say that the Gutman index has less predictive ability compared to degree-distance index. Further the correlation of Gutman index with physical properties of alkanes is very less and correlation coefficient value lies between 0.187 to 0.892.

The another degree-distance based topological index viz, Ashwini index. The predicting power of Ashwini index with physical properties of alkanes is too less. The correlation coefficient value of Ashwini index lies between 0.288 to 0.519.

Motivated by Gutman index and Ashwini index, Here we introduce a new degree-distance based topological invariant viz, SM Index. The QSPR study of SNM Index in table-9 shows good predicting power for alkanes.

From practical point of view, topological indices for which the absolute value of the correlation coefficient is less than 0.8 can be characterized useless. Thus the QSPR study of these distance and degree-distance based topological indices with physical properties of alkanes helps us to characterize useful topological indices indices with absolute values of correlation coefficients lies between 0.8 to 0.970.

References

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    A. T. Balban, Chemical applications of graph theory, Academic Press (1976).

  • [2]

    A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta. Appl. Math. 66 (2001) 211–249.

  • [3]

    H. Dong, X. Guo, Ordering trees by their Wiener indices, MATCH Commun. Math. Comput. Chem. 56 (2006) 527– 540.

  • [4]

    J. Devillers, A. T. Balban, Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach Science Publishers, Amsterdam, Netherlands, (1999).

  • [5]

    A. A. Dobrynin, A. A. Kochetova, Degree-distance of a graph: A degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci., 34 (1994) 1082–1086.

  • [6]

    X. Li, I. Gutman, Mathematical aspects of Randić-type molecular structure descriptors, Univ. Kragujevac, 2006.

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    I. Gutman, N. Trinajstić Graph theory and molecular orbitals. Total p-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538.

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    I. Gutman, B. Furtula, M. Petrović Terminal Wiener index, J. Math. Chem. 46 (2009) 522–531.

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    I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34(1994) 1087– 1089.

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    F. Harary, Graph Theory, Addison–Wesely, Reading, 1969.

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    B. Horvat, T. Pisanski, M. Randić Terminal polynomials and star-like graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 493–512.

  • [12]

    S. M. Hosamani, An improved proof for the Wiener index when diam G 2, Math. Sci. Lett. 5(2)(2016) 1–2.

  • [13]

    S. M. Hosamani, Ashwini index of a graph, Int. J. Industrial Mathematics, 8(4)(2016) 377–384.

  • [14]

    G. Liu, Z. Jia,W. Gao, (2018). Ontology Similarity Computing Based on Stochastic Primal Dual Coordinate Technique. Open j. math. sci., 2(1), 221-227.

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    D. Plavsić, S. Nikolić, N. Trinajstić, On the Harary index for the characterization of chemical graphs, J. Math. Chem 12(1993) 235–250.

  • [16]

    M. Randić, J. Zupan, Highly compact 2-D graphical representation of DNA sequences, SAR QSAR Environ. Res. 15 (2004) 191–205.

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    M. Randić, J. Zupan, D. Vikić Topić, On representation of proteins by star-like graphs, J. Mol. Graph. Modell. 26(2007) 290–305.

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    M. Randićc, Quantitative Structure- Property Relationship: boiling points and planar benzenoids, New. J. Chem. 20 (1996) 1001–1009.

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    M. Randić, Novel molecular descriptor for structureroperty studies, Chem. Phys.Lett. 211 (1993) 478–483.

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    M. Randić, Comparative structure-property studies: Regressions using a single descriptor. Croat. Chem. Acta 66 (1993) 289–312.

  • [21]

    M. Randić, On characterization of molecular branching. J. Am. Chem. Soc. 97 (1975) 6609–6615.

  • [22]

    M. Randić, M. Pompe, On characterization of CC double bond in alkenes, SAR and QSAR Environ. Res. 10 (1999) 451–471.

  • [23]

    L. A. Szkely, H. Wang, T. Wu, The sum of distances between the leaves of a tree and the semi-regular property, Discr. Math. 311 (2011) 1197–1203.

  • [24]

    Z. Tang, L. Liang, W. Gao, (2018). Wiener polarity index of quasi-tree molecular structures. Open j. math. sci., 2(1), 73-83.

  • [25]

    N. Trinajstić, Chemical graph theory, CRC Press (1992).

  • [26]

    H. Wiener Structural determination of parafin boiling points, Journal of the American Chemical Society, 1(69) (1947) 17-20.

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    L. Yan, M. R. Farahani, W. Gao, (2018). Distance-based Indices Computation of Symmetry Molecular Structures. Open j. math. sci., 2(1), 323-337.

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  • [1]

    A. T. Balban, Chemical applications of graph theory, Academic Press (1976).

  • [2]

    A. A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: theory and applications, Acta. Appl. Math. 66 (2001) 211–249.

  • [3]

    H. Dong, X. Guo, Ordering trees by their Wiener indices, MATCH Commun. Math. Comput. Chem. 56 (2006) 527– 540.

  • [4]

    J. Devillers, A. T. Balban, Topological indices and related descriptors in QSAR and QSPR, Gordon and Breach Science Publishers, Amsterdam, Netherlands, (1999).

  • [5]

    A. A. Dobrynin, A. A. Kochetova, Degree-distance of a graph: A degree analogue of the Wiener index, J. Chem. Inf. Comput. Sci., 34 (1994) 1082–1086.

  • [6]

    X. Li, I. Gutman, Mathematical aspects of Randić-type molecular structure descriptors, Univ. Kragujevac, 2006.

  • [7]

    I. Gutman, N. Trinajstić Graph theory and molecular orbitals. Total p-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (1972), 535–538.

  • [8]

    I. Gutman, B. Furtula, M. Petrović Terminal Wiener index, J. Math. Chem. 46 (2009) 522–531.

  • [9]

    I. Gutman, Selected properties of the Schultz molecular topological index, J. Chem. Inf. Comput. Sci. 34(1994) 1087– 1089.

  • [10]

    F. Harary, Graph Theory, Addison–Wesely, Reading, 1969.

  • [11]

    B. Horvat, T. Pisanski, M. Randić Terminal polynomials and star-like graphs, MATCH Commun. Math. Comput. Chem. 60 (2008) 493–512.

  • [12]

    S. M. Hosamani, An improved proof for the Wiener index when diam G 2, Math. Sci. Lett. 5(2)(2016) 1–2.

  • [13]

    S. M. Hosamani, Ashwini index of a graph, Int. J. Industrial Mathematics, 8(4)(2016) 377–384.

  • [14]

    G. Liu, Z. Jia,W. Gao, (2018). Ontology Similarity Computing Based on Stochastic Primal Dual Coordinate Technique. Open j. math. sci., 2(1), 221-227.

  • [15]

    D. Plavsić, S. Nikolić, N. Trinajstić, On the Harary index for the characterization of chemical graphs, J. Math. Chem 12(1993) 235–250.

  • [16]

    M. Randić, J. Zupan, Highly compact 2-D graphical representation of DNA sequences, SAR QSAR Environ. Res. 15 (2004) 191–205.

  • [17]

    M. Randić, J. Zupan, D. Vikić Topić, On representation of proteins by star-like graphs, J. Mol. Graph. Modell. 26(2007) 290–305.

  • [18]

    M. Randićc, Quantitative Structure- Property Relationship: boiling points and planar benzenoids, New. J. Chem. 20 (1996) 1001–1009.

  • [19]

    M. Randić, Novel molecular descriptor for structureroperty studies, Chem. Phys.Lett. 211 (1993) 478–483.

  • [20]

    M. Randić, Comparative structure-property studies: Regressions using a single descriptor. Croat. Chem. Acta 66 (1993) 289–312.

  • [21]

    M. Randić, On characterization of molecular branching. J. Am. Chem. Soc. 97 (1975) 6609–6615.

  • [22]

    M. Randić, M. Pompe, On characterization of CC double bond in alkenes, SAR and QSAR Environ. Res. 10 (1999) 451–471.

  • [23]

    L. A. Szkely, H. Wang, T. Wu, The sum of distances between the leaves of a tree and the semi-regular property, Discr. Math. 311 (2011) 1197–1203.

  • [24]

    Z. Tang, L. Liang, W. Gao, (2018). Wiener polarity index of quasi-tree molecular structures. Open j. math. sci., 2(1), 73-83.

  • [25]

    N. Trinajstić, Chemical graph theory, CRC Press (1992).

  • [26]

    H. Wiener Structural determination of parafin boiling points, Journal of the American Chemical Society, 1(69) (1947) 17-20.

  • [27]

    L. Yan, M. R. Farahani, W. Gao, (2018). Distance-based Indices Computation of Symmetry Molecular Structures. Open j. math. sci., 2(1), 323-337.